Spectral function expressions

We demonstrate that the spectral function of valence harmonic vibrations of a diatomic group that effects rotational reorientations is broadened by w. The vector of atom C displacements relative to the atom B (see Fig. A2.1) may be represented as x(t)e(t), where x(t) is the change in the length of the valence bond oriented at the time t along the unit vector e(/). Characteristic periods of valence vibrations are much shorter than periods of changes in unit vector orientations. As a consequence, the GF of the displacements defined by Eq. (4.2.1) can be expressed approximately as ... 

The spectral function thus has a Lorentz shape with a halfwidth at the half distribution height equal to the average reorientation frequency w. If expressed in spectroscopic units (cm 1), the halfwidth Avv2 amounts to cd2nca (c0 designates the velocity of light in vacuo). 

Expressions for the medium modifications of the cluster distribution functions can be derived in a quantum statistical approach to the few-body states, starting from a Hamiltonian describing the nucleon-nucleon interaction by the potential V"(12, l/2/) (1 denoting momentum, spin and isospin). We first discuss the two-particle correlations which have been considered extensively in the literature [5,7], Results for different quantities such as the spectral function, the deuteron binding energy and wave function as well as the two-nucleon scattering phase shifts in the isospin singlet and triplet channel have been evaluated for different temperatures and densities. The composition as well as the phase instability was calculated. 

Using equation (12) we can express the diagonal elements of the spectral function and calculate the local DOS at site i ... 

A better approach actually exists. There is considerable empirical and theoretical evidence that the various spectral functions, g(v), of binary systems are indeed closely modeled by a combination of the BC and K0 profiles, Eqs. 5.105 and 5.108. These are functionals of reduced spectral moments, Mo, Mi/Mo, and M2/M0, which in the classical limit may be expressed in terms of reduced temperature, to the extent that interaction potentials are describable by reduced potentials. 

In these formulae, Gd is the desymmetrized profile, Gc is the classical (symmetric) line profile c and T are angular frequency and temperature. In all cases, the desymmetrized function Go(ft>) obeys Eq. 5.73 exactly. We note that at low frequencies, h(o < kT, the four expressions are practically equivalent. However, at high frequencies the results of these desymmetrizations differ strikingly. One needs only to compare the magnitude of the factors of Gc of the upper three defining equations, for ft) — +oo, to realize enormous differences among these. Hence, the question arises as to which one (if any) of these procedures approximates the exact quantum profile, G(co). We note that in EgelstafFs procedure the desymmetrization is accomplished in the time domain rather than the frequency domain. The classical correlation function, Cci(t), and spectral function, Gci (co), are related by Fourier transform. 

The induced dipole moment p is expressed in the form of Eq. 4.18 in spherical components [314, 317]. As was seen in [43], we obtain the spectral function as a multiple sum of incoherent components,... 

When Z is sufficiently large, it is convenient to replace the sums involved in the above expressions by integrals and define continuous distributions of relaxation and retardation times. These spectral functions are usually defined on log time scales through the relations (9) ... 

A. Spectral Function and Complex Susceptibility (General Expressions)... 

Expression of Complex Susceptibility Through Spectral Function... 

E. Summation of Series in Expression for the Spectral Function IV. Hat-Flat Model... 

Using the so-called planar libration-regular precession (PL-RP) approximation, it is possible to reduce the double integral for the spectral function to a simple integral. The interval of integration is divided in the latter by two intervals, and in each one the integrands are substantially simplified. This simplification is shown to hold, if a qualitative absorption frequency dependence should be obtained. Useful simple formulas are derived for a few statistical parameters of the model expressed in terms of the cone angle (5 and of the lifetime x. A small (3 approximation is also considered, which presents a basis for the hybrid model. The latter is employed in Sections IV and VIII, as well as in other publications (VIG). 

We derive an analytical expression for the spectral function in terms of a double integral, which differs from the formulas given in Section III by account of finiteness of the well depth. Two important approximations are also given, in which the spectral function is represented by simple integrals. These... 

Third, the expression for the spectral function pertinent to the HO model is derived in detail using the ACF method. Some general results given in GT and VIG (and also in Section II) are confirmed by calculations, in which an undamped harmonic law of motion of the bounded charged particles is used explicitly. The complex susceptibility, depending on a type of a collision model,... 

In the second period, which was ended by review GT after the average perturbation theorem was proved, it became possible to get the Kubo-like expression for the spectral function L(z) (GT, p. 150). This expression is applicable to any axially symmetric potential well. Several collision models were also considered, and the susceptibility was expressed through the same spectral function L(z) (GT, p. 188). The law of motion of the particles should now be determined only by the steady state. So, calculations became much simpler than in the period (1). The best achievements of the period (2) concern the cone-confined rotator model (GT, p. 231), in which the dipoles were assumed to librate in space in an infinitely deep rectangular well, and applications of the theory to nonassociated liquids (GT, p. 329). 

The spectral function (22a) is a main object of further calculations given in Sections III-VI and VII-IX. In the next subsection we shall derive a more convenient expression for L(z). 

We shall get from (70a) and (70b) another formula for the spectral function L(z). At first, we express the series S(k) through elementary functions (the derivation is given in Section III.E) ... 

We have obtained the expression given in GT, p. 225 for the spectral function of free rotors moving in a homogeneous potential in the interval between strong collisions see also VIG, Eqs. (7.12) and (7.13). So, the subscript F means free. The subscript R in Eq. (74c) is used as an initial letter of restriction. Indeed, as it follows from the comparison of Eq. (77) with Eq. (74a), the second term of the last equation expresses the steric-restriction effect arising for free rotation due to a potential wall. If we set, for example, p = 7t, what corresponds to a complete rotation (without restriction) of a dipole-moment vector p, then we find from Eqs. (74a)-(74c) that LR z) = 0 and L(z) = Lj,(z). This result confirms our statement about restriction. ... 

Then we divide the interval of integration over / by two above-mentioned intervals. We retain rigorous expression in exponential and power functions involved in the integrand in Eqs. (70a) and (70b), but in other co-factors we set / = 0 and g = 0 in the first and second intervals, respectively. The spectral function then is approximated by a sum... 

We modify Eqs. (79) and (80) in the fashion described in Section IV.B.l. Then we have the following expression for the librational spectral function ... 

We shall obtain here expression for the spectral function of the dipoles performing complete rotation by using an interpolation approximation, which allows us to ignore the distinction between hindered and free rotations. Both types of motion we mark by the same superscript °. 

If the angle (3 is much less than 1, then, in accord with Figs. 7 and 9, the most part of the rotators move freely under effect of a constant potential U0, since their trajectories do not intersect the conical cavity. A small part of the rotators moves along a trajectory of the type 1 shown in Fig. 10. However, at d > (3—that is, in the most part of such a trajectory—they are affected by the same constant potential U0- Therefore, for this second group of the particles the law of motion is also rather close to the law of free rotation. For the latter the dielectric response is described by Eq. (77). We shall represent this formula as a particular case of the general expression (51), in which the contributions to the spectral function due to longitudinal A) and transverse KL components are determined, respectively, by the first and second terms under summation sign. Free rotators present a medium isotropic in a local-order scale. Therefore, we set = K . Then the second term... 

Finally, a simplified expression for the spectral function of the rotators is given by... 

The phase regions occupied by the librators and precessors are depicted in Fig. 27 in coordinates h, l2 for the parameters u = 5.9, p = %/9. We take two values of the form factor/ 0.65 in (a) and 0.85 in (b). When/increases, the / and 2P areas extend to the larger h and l values. The values of Vmin are shown as functions of l in Fig. 21c. In this example (and in the calculations described in Section V.C) the potential well depth U0 is much greater than kBT, that is, u> 1. We see in Fig. 27 that for the J and 2P areas the boundary values for h are still greater than it. This property is used to simplify analytical expressions for the spectral functions. 

In accord with the planar libration approximation, we first come from representation of the spectral function for motion of a dipole in a plane, where integration over l is lacking by definition, so only integration over energy h is employed. We shall find in this way the function (203). As a next step we carry out integration over l, so that a rather simple expression (171) for the spectral function L(z) will be obtained. 

Let us calculate the broadband spectra of liquid water H20 and D20. The adopted experimental data are presented in Table XII. In accord with the scheme (238), we use Eq. (249) for the complex susceptibility x and use Eqs. (242) and (243) for the modified spectral function R(z). All other expressions used in these calculations are the same as were employed in Section V. 

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